Minimizing Monochromatic Subgraphs of $K_{n,n}$

Charles Gong

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arXiv · 2410.19076

Minimizing Monochromatic Subgraphs of $K_{n,n}$

Charles Gong

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Abstract

Given any $r$-edge coloring of $K_{n,n}$, how large is the maximum (over all $r$ colors) sized monochromatic subgraph guaranteed to be? We give answers to this problem for $r \leq 8$, when $r$ is a perfect square, and when $r$ is one less than a perfect square all up to a constant additive term that depends on $r$. We give a lower bound on this quantity that holds for all $r$ and is sharp when $r$ is a perfect square up to a constant additive term that depends on $r$. Finally, we give a construction for all $r$ which provides an upper bound on this quantity up to a constant additive term that depends on $r$, and which we conjecture is also a lower bound.

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BibTeX RIS

Charles Gong. 2024-10-24. Minimizing Monochromatic Subgraphs of $K_{n,n}$. https://arxiv.org/abs/2410.19076

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